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Random walk and fluctuation theory. (English) Zbl 0982.60038

Shanbhag, D. N. (ed.) et al., Stochastic processes: Theory and methods. Amsterdam: North-Holland/ Elsevier. Handb. Stat. 19, 171-213 (2001).
This is an extensive survey article on random walks. The author surveys and explains a host of classical and recent results, applications, the history of the research, and lists a comprehensive literature. Proofs or new results are not contained, but for every result there are precise references and interpretations provided, if possible. The long list of provided applications to real-world phenomena include gamble, electrostatics, mathematical finance, card shuffling, and electrical networks. The text is written in a rather understandable and pleasant way and gives the reader a quick and precise summary of the important results on random walks. It is amazing how many interesting and important aspects are collected, explained and put into relation to each other.
The text starts with the basic properties of simple random walk in the Euclidean space, like recurrence and transience, generating functions, Polyá’s theorem, (discrete) harmonic analysis, and (discrete) renewal theory and potential theory. Then it surveys the coupling method and Donsker’s invariance principle and discusses random walks on special structures, like groups, graphs and some more specified structures. It concisely mentions some variants of random walks like random walk in random environment, self-avoiding walk and branching random walk, and then turns to the second main topic, the fluctuation theory. Here special emphasis is put on Spitzer’s identity for the distribution maximum of a random walk after \(n\) steps, the Wiener-Hopf method, and Spitzer’s arc-sine law. Consequences and applications to ballot theorems, queues, barrier problems and non-parametric statistics are treated as well.
For the entire collection see [Zbl 0961.60001].
Reviewer: W.König (Berlin)

MSC:

60G50 Sums of independent random variables; random walks
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